How to translate and rotate the heading of landscaped pages?

The lscape package is not designed for this. It's designed for rotating wide figures or tables, for example. And the geometry package explicitly says that \newgeometry can't change the paper size or orientation. So I don't think there's a way to do this automatically.

You can include landscape oriented pdf pages using the pdfpages package. (Include them with the [landscape] option.)

A new solution

You could also use the textpos package to place the headers. By combining this with the fancyhdr package, you can pretty much automate it.

\documentclass[twoside]{book}
\usepackage[a5paper,hmargin=3cm,vmargin=5cm]{geometry}
\usepackage{lscape,lipsum,graphicx}
\usepackage[absolute]{textpos}
\usepackage{fancyhdr}

\fancypagestyle{lscape}{% 
\fancyhf{} % clear all header and footer fields 
\fancyfoot[LE]{%
\begin{textblock}{20}(1,5){\rotatebox{90}{\leftmark}}\end{textblock}
\begin{textblock}{1}(13,10.5){\rotatebox{90}{\thepage}}\end{textblock}}
\fancyfoot[LO] {%
\begin{textblock}{1}(13,10.5){\rotatebox{90}{\thepage}}\end{textblock}
\begin{textblock}{20}(1,13.25){\rotatebox{90}{\rightmark}}\end{textblock}}
\renewcommand{\headrulewidth}{0pt} 
\renewcommand{\footrulewidth}{0pt}}

\setlength{\TPHorizModule}{1cm}
\setlength{\TPVertModule}{1cm}

\begin{document}
\chapter{A chapter}
\section{A section}
\lipsum
\section{A section}
\lipsum

\newgeometry{hmargin=3cm,vmargin=5cm}
\thispagestyle{lscape}
\pagestyle{lscape}
\begin{landscape}
\lipsum[1-3]

\end{landscape}

\restoregeometry
\pagestyle{headings}
\lipsum[6]

\end{document}

I need page numbers on the landscaped pages. And the page numbers should be at the bottom as shown in the second figure above.

I have an idea. We can use \fancypagestyle to define a new pagestyle with landscaped page numbers. I use tikz to put the page number.

\fancypagestyle{lscapedplain}{%
  \fancyhf{}
  \fancyfoot{%
    \tikz[remember picture,overlay]
      \node[outer sep=1cm,above,rotate=90] at (current page.east) {\thepage};}
}

This would also be done by \newpagestyle from titlesec.

Analternative approach using package typearea to switch to landscape-mode mid-document. Though it is really strange to do that.

\documentclass[usegeometry,paper=a5,pagesize,headinclude]{book}
\usepackage{typearea}
\usepackage{lipsum}

\begin{document}
\chapter{One}
\lipsum[1]

\section{One One}
\lipsum[2]

\storeareas\normalsetting
\KOMAoption{paper}{landscape}
\areaset{2\textwidth}{.9\textheight}
\recalctypearea
    \lipsum[3-5]
\clearpage
\normalsetting
\lipsum[6]

\end{document}